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<div  id='write'  class = 'is-mac first-line-indent'><h2><a name="函数极限的定理证明和例题" class="md-header-anchor"></a><span>函数极限的定理证明和例题</span></h2><blockquote><p><span>张一极</span></p><p><span>2020-0514-16:41</span></p></blockquote><h2><a name="1定理" class="md-header-anchor"></a><span>1.定理:</span></h2><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n7" cid="n7" mdtype="math_block">
			
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y="-213"></use></g><use xlink:href="#E508-MJMAIN-7C" x="3097" y="0"></use></g><use xlink:href="#E508-MJMAIN-3C" x="14534" y="0"></use><use xlink:href="#E508-MJMATHI-3B4" x="15590" y="0"></use><g transform="translate(16041,0)"><g transform="translate(250,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(58.579) matrix(1 0 0 -1 0 0)">时</text></g></g><use xlink:href="#E508-MJMAIN-2C" x="17300" y="0"></use><g transform="translate(17745,0)"><g transform="translate(250,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(58.579) matrix(1 0 0 -1 0 0)">有</text></g></g><use xlink:href="#E508-MJMATHI-66" x="19005" y="0"></use><use xlink:href="#E508-MJMAIN-28" x="19555" y="0"></use><use xlink:href="#E508-MJMATHI-78" x="19944" y="0"></use><use xlink:href="#E508-MJMAIN-29" x="20516" y="0"></use><use xlink:href="#E508-MJMAIN-3E" x="21183" y="0"></use><use xlink:href="#E508-MJMAIN-30" x="22238" y="0"></use><use xlink:href="#E508-MJMAIN-28" x="23738" y="0"></use><g transform="translate(24127,0)"><g transform="translate(250,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(58.579) matrix(1 0 0 -1 0 0)">或</text></g></g><use xlink:href="#E508-MJMATHI-66" x="25387" y="0"></use><use xlink:href="#E508-MJMAIN-28" x="25937" y="0"></use><use xlink:href="#E508-MJMATHI-78" x="26326" y="0"></use><use xlink:href="#E508-MJMAIN-29" x="26898" y="0"></use><use xlink:href="#E508-MJMAIN-3C" x="27565" y="0"></use><use xlink:href="#E508-MJMAIN-30" x="28621" y="0"></use><use xlink:href="#E508-MJMAIN-29" x="29121" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-441">\begin{array}{c}\text { 定理 } \mathbf{1}(\text { 函数极 } \text { 限的唯一性 }) \quad \text { 如果 } \lim _{x \rightarrow x_{0}} f(x) \text { 存在 }, \text { 那么这极 } \text { 限唯一 } \\ \text { 定理 } \mathbf{2}(\text { 函数极限的局部有界性 }) \quad \text { 如果 } \lim _{x \rightarrow 0} f(x)=A, \text { 那 } \text { 么存在常数 } M>0 \text { 和 } \delta \\ >0, \text { 使得当 } 0<\left|x-x_{0}\right|<\delta \text { 时 }, \text { 有 }|f(x)| \leqslant M \\ \text { 证 } \quad \text { 因为 } \lim _{x \rightarrow x_{0}} f(x)=A, \text { 所以取 } \varepsilon=1, \text { 则 } \exists \delta>0, \text { 当 } 0<\left|x-x_{0}\right|<\delta \text { 时,有 } \\ \qquad \begin{array}{c}|f(x)-A|<1 \Rightarrow|f(x)| \leqslant|f(x)-A|+|A|<|A|+1 \\ \text { 记 } M=|A|+1, \text { 则定理 } 2 \text { 就获得证明. } \\ \text { 定理 } \mathbf{3}(\text { 函数极 } \text { 限的局部保号性 }) \quad \text { 如果 } \lim _{x \rightarrow x_{0}} f(x)=A, \text { 且 } A>0 \quad(\text { 或 } A<0), \text { 那 } \text { 么 }\end{array} \\ \text { 存在常数 } \delta>0, \text { 使得当 } 0<\left|x-x_{0}\right|<\delta \text { 时 }, \text { 有 } f(x)>0 \quad(\text { 或 } f(x)<0)\end{array}</script>
					</div></div><h3><a name="证明函数局部保号性" class="md-header-anchor"></a><span>证明函数局部保号性:</span></h3><p><span>首先,fx的极限A&gt;0</span></p><p><span>证明:</span></p><p><span>任取</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="49.109ex" height="3.002ex" viewBox="0 -880.7 21144.2 1292.7" role="img" focusable="false" style="vertical-align: -0.957ex;"><defs><path stroke-width="0" id="E137-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path><path stroke-width="0" id="E137-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E137-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E137-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E137-MJMAIN-2203" d="M56 661T56 674T70 694H487Q497 686 500 679V15Q497 10 487 1L279 0H70Q56 7 56 20T70 40H460V327H84Q70 334 70 347T84 367H460V654H70Q56 661 56 674Z"></path><path stroke-width="0" id="E137-MJMATHI-3B4" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z"></path><path stroke-width="0" id="E137-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path stroke-width="0" id="E137-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E137-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E137-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E137-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="0" id="E137-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E137-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E137-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E137-MJMATHI-3B5" x="0" y="0"></use><use xlink:href="#E137-MJMAIN-3E" x="743" y="0"></use><use xlink:href="#E137-MJMAIN-30" x="1799" y="0"></use><use xlink:href="#E137-MJMAIN-2C" x="2299" y="0"></use><use xlink:href="#E137-MJMAIN-2203" x="2744" y="0"></use><use xlink:href="#E137-MJMATHI-3B4" x="3300" y="0"></use><use xlink:href="#E137-MJMAIN-3E" x="4029" y="0"></use><use xlink:href="#E137-MJMAIN-30" x="5084" y="0"></use><use xlink:href="#E137-MJMAIN-2C" x="5584" y="0"></use><g transform="translate(6029,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(58.579) matrix(1 0 0 -1 0 0)">都</text></g><g transform="translate(6753,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(58.579) matrix(1 0 0 -1 0 0)">有</text></g><use xlink:href="#E137-MJMAIN-30" x="7477" y="0"></use><use xlink:href="#E137-MJMAIN-3C" x="8255" y="0"></use><use xlink:href="#E137-MJMAIN-7C" x="9310" y="0"></use><use xlink:href="#E137-MJMATHI-78" x="9588" y="0"></use><use xlink:href="#E137-MJMAIN-2212" x="10383" y="0"></use><g transform="translate(11383,0)"><use xlink:href="#E137-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E137-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E137-MJMAIN-7C" x="12408" y="0"></use><use xlink:href="#E137-MJMAIN-3C" x="12964" y="0"></use><use xlink:href="#E137-MJMATHI-3B4" x="14020" y="0"></use><use xlink:href="#E137-MJMAIN-2C" x="14471" y="0"></use><use xlink:href="#E137-MJMAIN-7C" x="14916" y="0"></use><use xlink:href="#E137-MJMATHI-66" x="15194" y="0"></use><use xlink:href="#E137-MJMAIN-28" x="15744" y="0"></use><use xlink:href="#E137-MJMATHI-78" x="16133" y="0"></use><use xlink:href="#E137-MJMAIN-29" x="16705" y="0"></use><use xlink:href="#E137-MJMAIN-2212" x="17316" y="0"></use><use xlink:href="#E137-MJMATHI-41" x="18316" y="0"></use><use xlink:href="#E137-MJMAIN-7C" x="19066" y="0"></use><use xlink:href="#E137-MJMAIN-3C" x="19622" y="0"></use><use xlink:href="#E137-MJMATHI-3B5" x="20678" y="0"></use></g></svg></span><script type="math/tex">\varepsilon>0, \exists\delta>0,都有0<|x-x_0|<\delta,|f(x)-A|<\varepsilon</script></p><p><span>那么我们可以通过这个式子,得到</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.935ex" height="2.73ex" viewBox="0 -822.1 9444 1175.6" role="img" focusable="false" style="vertical-align: -0.821ex;"><defs><path stroke-width="0" id="E174-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E174-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E174-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path><path stroke-width="0" id="E174-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path stroke-width="0" id="E174-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="0" id="E174-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E174-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E174-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E174-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E174-MJMATHI-41" x="0" y="0"></use><use xlink:href="#E174-MJMAIN-2212" x="972" y="0"></use><use xlink:href="#E174-MJMATHI-3B5" x="1972" y="0"></use><use xlink:href="#E174-MJMAIN-3C" x="2716" y="0"></use><use xlink:href="#E174-MJMATHI-66" x="3772" y="0"></use><use xlink:href="#E174-MJMAIN-28" x="4322" y="0"></use><use xlink:href="#E174-MJMATHI-78" x="4711" y="0"></use><use xlink:href="#E174-MJMAIN-29" x="5283" y="0"></use><use xlink:href="#E174-MJMAIN-3C" x="5949" y="0"></use><use xlink:href="#E174-MJMATHI-41" x="7005" y="0"></use><use xlink:href="#E174-MJMAIN-2B" x="7977" y="0"></use><use xlink:href="#E174-MJMATHI-3B5" x="8978" y="0"></use></g></svg></span><script type="math/tex">A-\varepsilon<f(x)<A+\varepsilon</script></p><p><span>既然极限存在且为A,那么可以任取一个</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.247ex" height="3.411ex" viewBox="0 -997.8 2689.9 1468.5" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E199-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path><path stroke-width="0" id="E199-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E199-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E199-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E199-MJMATHI-3B5" x="0" y="0"></use><use xlink:href="#E199-MJMAIN-3D" x="743" y="0"></use><g transform="translate(1521,0)"><g transform="translate(397,0)"><rect stroke="none" width="650" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E199-MJMATHI-41" x="84" y="593"></use><use transform="scale(0.707)" xlink:href="#E199-MJMAIN-32" x="209" y="-553"></use></g></g></g></svg></span><script type="math/tex">\varepsilon=\frac{A}{2}</script><span>,可得:</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.075ex" height="3.411ex" viewBox="0 -997.8 6921 1468.5" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E229-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E229-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E229-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path stroke-width="0" id="E229-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="0" id="E229-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E229-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E229-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E229-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="650" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E229-MJMATHI-41" x="84" y="593"></use><use transform="scale(0.707)" xlink:href="#E229-MJMAIN-32" x="209" y="-553"></use></g><use xlink:href="#E229-MJMAIN-3C" x="1168" y="0"></use><use xlink:href="#E229-MJMATHI-66" x="2223" y="0"></use><use xlink:href="#E229-MJMAIN-28" x="2773" y="0"></use><use xlink:href="#E229-MJMATHI-78" x="3162" y="0"></use><use xlink:href="#E229-MJMAIN-29" x="3734" y="0"></use><use xlink:href="#E229-MJMAIN-3C" x="4401" y="0"></use><g transform="translate(5179,0)"><g transform="translate(397,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E229-MJMAIN-33" x="84" y="615"></use><use transform="scale(0.707)" xlink:href="#E229-MJMAIN-32" x="84" y="-553"></use></g></g><use xlink:href="#E229-MJMATHI-41" x="6170" y="0"></use></g></svg></span><script type="math/tex">\frac{A}{2}<f(x)<\frac{3}{2}A</script><span>,又因为</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.36ex" height="3.411ex" viewBox="0 -997.8 5752.1 1468.5" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E252-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E252-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E252-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E252-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E252-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E252-MJMATHI-41" x="0" y="0"></use><use xlink:href="#E252-MJMAIN-3E" x="1027" y="0"></use><use xlink:href="#E252-MJMAIN-30" x="2083" y="0"></use><use xlink:href="#E252-MJMAIN-2C" x="2583" y="0"></use><g transform="translate(2861,0)"><g transform="translate(286,0)"><rect stroke="none" width="650" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E252-MJMATHI-41" x="84" y="593"></use><use transform="scale(0.707)" xlink:href="#E252-MJMAIN-32" x="209" y="-553"></use></g></g><use xlink:href="#E252-MJMAIN-3E" x="4196" y="0"></use><use xlink:href="#E252-MJMAIN-30" x="5252" y="0"></use></g></svg></span><script type="math/tex">A>0,\frac{A}{2}>0</script><span>,所以</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.672ex" height="2.73ex" viewBox="0 -822.1 3733.6 1175.6" role="img" focusable="false" style="vertical-align: -0.821ex;"><defs><path stroke-width="0" id="E258-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="0" id="E258-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E258-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E258-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E258-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E258-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E258-MJMATHI-66" x="0" y="0"></use><use xlink:href="#E258-MJMAIN-28" x="550" y="0"></use><use xlink:href="#E258-MJMATHI-78" x="939" y="0"></use><use xlink:href="#E258-MJMAIN-29" x="1511" y="0"></use><use xlink:href="#E258-MJMAIN-3E" x="2177" y="0"></use><use xlink:href="#E258-MJMAIN-30" x="3233" y="0"></use></g></svg></span><script type="math/tex">f(x)>0</script><span>. </span></p><p><span>证毕.</span></p><h2><a name="2函数和导函数唯一有界性的正确结论" class="md-header-anchor"></a><span>2.函数和导函数唯一有界性的正确结论: </span></h2><h3><a name="证明fx在ab有界则原函数在ab依然有界" class="md-header-anchor"></a><span>	</span><span>证明</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.129ex" height="2.967ex" viewBox="0 -957.1 2208.2 1277.5" role="img" focusable="false" style="vertical-align: -0.744ex;"><defs><path stroke-width="0" id="E360-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="0" id="E360-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E360-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E360-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E360-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E360-MJMATHI-66" x="0" y="0"></use><use xlink:href="#E360-MJMAIN-28" x="550" y="0"></use><use xlink:href="#E360-MJMATHI-78" x="939" y="0"></use><g transform="translate(1511,0)"><use xlink:href="#E360-MJMAIN-29" x="0" y="0"></use><g transform="translate(389,362)"><use transform="scale(0.5)" xlink:href="#E360-MJMAIN-2032" x="0" y="513"></use></g></g></g></svg></span><script type="math/tex">f(x)^{'}</script><span>在[a,b]有界,则原函数在[a,b]依然有界:</span></h3><p><span>根据拉格朗日中值定理:</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.059ex" height="2.65ex" viewBox="0 -820.7 11219.9 1141" role="img" focusable="false" style="vertical-align: -0.744ex;"><defs><path stroke-width="0" id="E369-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 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750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E369-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E369-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E369-MJMAIN-3D" d="M56 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x="0" y="0"></use><use xlink:href="#E369-MJMAIN-28" x="550" y="0"></use><use xlink:href="#E369-MJMATHI-62" x="939" y="0"></use><use xlink:href="#E369-MJMAIN-29" x="1368" y="0"></use><use xlink:href="#E369-MJMAIN-2212" x="1979" y="0"></use><use xlink:href="#E369-MJMATHI-66" x="2979" y="0"></use><use xlink:href="#E369-MJMAIN-28" x="3529" y="0"></use><use xlink:href="#E369-MJMATHI-61" x="3918" y="0"></use><use xlink:href="#E369-MJMAIN-29" x="4447" y="0"></use><use xlink:href="#E369-MJMAIN-3D" x="5114" y="0"></use><g transform="translate(6169,0)"><use xlink:href="#E369-MJMATHI-66" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E369-MJMAIN-2032" x="803" y="513"></use></g><use xlink:href="#E369-MJMAIN-28" x="7032" y="0"></use><use xlink:href="#E369-MJMATHI-3B4" x="7421" y="0"></use><use xlink:href="#E369-MJMAIN-29" x="7872" y="0"></use><use xlink:href="#E369-MJMAIN-28" x="8261" y="0"></use><use xlink:href="#E369-MJMATHI-62" x="8650" y="0"></use><use xlink:href="#E369-MJMAIN-2212" x="9301" y="0"></use><use xlink:href="#E369-MJMATHI-61" x="10301" y="0"></use><use xlink:href="#E369-MJMAIN-29" x="10830" y="0"></use></g></svg></span><script type="math/tex">f(b)-f(a)=f^{\prime}(\delta)(b-a)</script><span>可以将b定为变量,</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="1.699ex" viewBox="0 -502.3 1025.6 731.7" role="img" focusable="false" style="vertical-align: -0.533ex;"><defs><path stroke-width="0" id="E374-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E374-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E374-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E374-MJMAIN-30" x="808" y="-213"></use></g></svg></span><script type="math/tex">x_0</script><span>作为一个值</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="32.417ex" height="2.967ex" viewBox="0 -957.1 13957.3 1277.5" role="img" focusable="false" style="vertical-align: -0.744ex;"><defs><path stroke-width="0" id="E444-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="0" id="E444-MJMAIN-28" d="M94 250Q94 319 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0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E444-MJMATHI-66" x="0" y="0"></use><use xlink:href="#E444-MJMAIN-28" x="550" y="0"></use><use xlink:href="#E444-MJMATHI-78" x="939" y="0"></use><use xlink:href="#E444-MJMAIN-29" x="1511" y="0"></use><use xlink:href="#E444-MJMAIN-2212" x="2122" y="0"></use><use xlink:href="#E444-MJMATHI-66" x="3122" y="0"></use><g transform="translate(3839,0)"><use xlink:href="#E444-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E444-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E444-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E444-MJMAIN-29" x="1414" y="0"></use></g><use xlink:href="#E444-MJMAIN-3D" x="5920" y="0"></use><g transform="translate(6976,0)"><use xlink:href="#E444-MJMATHI-66" x="0" y="0"></use><g transform="translate(568,362)"><use transform="scale(0.5)" xlink:href="#E444-MJMAIN-2032" x="0" y="513"></use></g></g><use xlink:href="#E444-MJMAIN-28" x="7852" y="0"></use><use xlink:href="#E444-MJMATHI-3B4" x="8241" y="0"></use><use xlink:href="#E444-MJMAIN-29" x="8692" y="0"></use><use xlink:href="#E444-MJMAIN-28" x="9081" y="0"></use><use xlink:href="#E444-MJMATHI-78" x="9470" y="0"></use><use xlink:href="#E444-MJMAIN-2212" x="10264" y="0"></use><g transform="translate(11264,0)"><use xlink:href="#E444-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E444-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E444-MJMAIN-29" x="12290" y="0"></use><use xlink:href="#E444-MJMAIN-2192" x="12957" y="0"></use></g></svg></span><script type="math/tex"> f(x)-f\left(x_{0}\right)=f^{'}(\delta)(x-x_{0})\to</script></p><p><span>		</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="32.774ex" height="2.65ex" 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y="0"></use><use transform="scale(0.707)" xlink:href="#E437-MJMAIN-2032" x="803" y="513"></use></g><use xlink:href="#E437-MJMAIN-28" x="8672" y="0"></use><use xlink:href="#E437-MJMATHI-3B4" x="9061" y="0"></use><use xlink:href="#E437-MJMAIN-29" x="9512" y="0"></use><g transform="translate(10068,0)"><use xlink:href="#E437-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E437-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E437-MJMAIN-2212" x="1183" y="0"></use><g transform="translate(2183,0)"><use xlink:href="#E437-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E437-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E437-MJMAIN-29" x="3208" y="0"></use></g><use xlink:href="#E437-MJMAIN-7C" x="13833" y="0"></use></g></svg></span><script type="math/tex">|f(x)|=| f\left(x_{0}\right) + f^{\prime}(\delta)\left(x-x_{0}\right) |</script></p><p><span>由于绝对值不等式|a+b|&lt;=|a|+|b|可得</span></p><p><span>		</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 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y="513"></use></g><use xlink:href="#E476-MJMAIN-28" x="21828" y="0"></use><use xlink:href="#E476-MJMATHI-3B4" x="22217" y="0"></use><use xlink:href="#E476-MJMAIN-29" x="22668" y="0"></use><use xlink:href="#E476-MJMAIN-7C" x="23057" y="0"></use><use xlink:href="#E476-MJMAIN-7C" x="23335" y="0"></use><g transform="translate(23780,0)"><use xlink:href="#E476-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E476-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E476-MJMAIN-2212" x="1183" y="0"></use><g transform="translate(2183,0)"><use xlink:href="#E476-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E476-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E476-MJMAIN-29" x="3208" y="0"></use></g><use xlink:href="#E476-MJMAIN-7C" x="27544" y="0"></use></g></svg></span><script type="math/tex">|f(x)|=| f\left(x_{0}\right) + f^{\prime}(\delta)\left(x-x_{0}\right) |<=| f\left(x_{0}\right)| + |f^{\prime}(\delta)||\left(x-x_{0}\right)|</script><span> 其中导函数有界,</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.55ex" height="2.016ex" viewBox="0 -638.7 2820 868.1" role="img" focusable="false" style="vertical-align: -0.533ex;"><defs><path stroke-width="0" id="E481-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path 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y="0"></use><use transform="scale(0.707)" xlink:href="#E509-MJMAIN-28" x="3200" y="0"></use><use transform="scale(0.707)" xlink:href="#E509-MJMATHI-78" x="3589" y="0"></use><use transform="scale(0.707)" xlink:href="#E509-MJMAIN-2212" x="4161" y="0"></use><use transform="scale(0.707)" xlink:href="#E509-MJMAIN-32" x="4939" y="0"></use><g transform="translate(3845,0)"><use transform="scale(0.707)" xlink:href="#E509-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E509-MJMAIN-32" x="550" y="674"></use></g></g></g></g></g></svg></span><script type="math/tex">f(x)=\frac{|x| \sin (x-2)}{x(x-1)(x-2)^{2}}</script><span>的有界区间:(-1,0),(0,1),(1,2),(2,3)</span></p><p><span>闭区间的连续函数有界,如果是开区间,需要增加定义的是两端有界,中间闭区间连续,则可以证明函数在开区间内有界</span></p><p><span>所以本题的四个区间,分界点分别是0,1,2,依次写出从不同方向趋近于0,1,2的函数极限值,可得知</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="30.018ex" height="4.907ex" 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y="0"></use><use transform="scale(0.707)" xlink:href="#E691-MJMAIN-6E" x="672" y="0"></use><use transform="scale(0.707)" xlink:href="#E691-MJMAIN-33" x="1463" y="0"></use></g><g transform="translate(400,-391)"><use transform="scale(0.707)" xlink:href="#E691-MJMAIN-31"></use><use transform="scale(0.707)" xlink:href="#E691-MJMAIN-38" x="500" y="0"></use></g></g></g></g></g></g></g></svg></span><script type="math/tex">lim_{x\to-1^{+}}\frac{(x) \sin (x-2)}{x (x-1)(x-2)^{2}}\\ \begin{array}{l}=(-1) \frac{\sin (-3)}{(-2) \cdot 9} \\ =-\frac{\sin 3}{18}\end{array}</script></p><p><span>因两侧极限存在且函数连续,则在(-1,0)区间内函数有界.命题得证</span></p></div>
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